Because of the spatial dependence of the local-density and gradient coefficients brought by the temperature variations, finding an analytical solution to Eq. (REF ) is in general a hard problem. Here, we will limit our discussion to a linear temperature profile along \(x\) , i.e., \(T(x)=T^0+\alpha x\) , with \(T^0\)  a reference temperature and \(\alpha \)  the temperature gradient. In turn, this allows us to linearize the coefficients \(A^T_{QQ}\)  and \(B^T_{QQ}\)  about \(T^0\) , i.e., \(A^T_{QQ}\approx A^0_{QQ}+\tfrac{\partial A_{QQ}}{\partial T}\big |_{0}\alpha x\)  and \(B^T_{QQ}\approx B^0_{QQ}+\tfrac{\partial B_{QQ}}{\partial T}\big |_{0}\alpha x\) .
As detailed in [1]}, the reference values of \(A^0_{QQ}\)  and \(B^0_{QQ}\)  are defined from the fourth- and second-order momenta of the direct correlation function of the reference system at \((T^0,\bar{\rho }_i)\) . In this work, these are computed using the hypernetted chain (HNC) approximation [2]}, which is known to provide a particularly good description of ionic fluids outside the region of the phase diagram delimited by the spinodal line [3]}, [4]}, [5]}.
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